Kathleen Alligood, Tim Sauer and James Yorke's Chaos is a well-written
book that provides a detailed introduction to dynamical systems
theory, with a strong emphasis on dissipative systems and
low-dimensional chaotic dynamics. The authors are mathematicians and
well known to the nonlinear dynamics community; in particular, Yorke
has for more than 20 years played a seminal role in the development of
chaos theory.

As might be expected from such a trio, the book
presents the mathematical theory with care. The presentation
nonetheless retains an introductory flavor throughout, by relying on
examples to convey the main ideas. Theorems are stated and often
proved, but in a reasonably humane manner: No effort is made to
formulate results in their most general setting, and the proofs are
carefully explained. On some points, such as the lambda lemma or the
differentiability of the stable manifold, where any proof is
necessarily rather technical, the authors provide references in which
the arguments can be found.

Although the widespread interest in
nonlinear dynamics has produced a number of books on the subject, very
few offer this level of mathematical detail and rigor packaged within
a genuinely introductory discussion. The organization of the material
has several notable characteristics: Overall, the subject is presented
in what I would call reverse order: The opening chapters treat chaos
in maps, fractal sets and various bifurcations (such as crises) that
arise in the examples; in later chapters, we find flows in one- and
two-dimensional phase spaces and a description of the elementary
bifurcations from fixed points. I would guess that this order might
be hard on some students, but I have not tried it in my own teaching.
It is undeniably economical, if you want to get to the sexy topics
fast.

While their main focus is on the mathematics and the study of
various model equations, the authors convey the relevance to real
experiments through 12 "lab visits" that appear as chapter appendices.
These are brief sketches of the appearance of the dynamical phenomena
when seen in the flesh, with examples chosen from chemistry, physics
and biology.

A second set of chapter appendices, written in the form
of extended homework exercises, expand on the theoretical development
of each chapter. These "challenges" include the proof of
Sharkovskii's theorem, the application of shadowing to justify
numerical computation of chaotic trajectories and the analysis of
synchronization between coupled chaotic systems.

In addition to
covering standard topics, also found in many other texts, there are
quite readable introductions to a number of advanced topics. In the
context of well-chosen examples, the authors describe Markov
partitions, invariant measures and natural measure, shadowing,
periodic orbit cascades and so-called Wada basins (whose fractal
boundaries have a structure that is almost beyond belief). Many of
these examples derive from the research of the authors.

The book
concludes with a chapter on phase-portrait reconstruction from
experimental time series and a discussion of various embedding
theorems. The book omits some things that I like to include in an
introductory graduate course. Surprisingly, the concept of structural
stability is not developed. As a result, although the pages
are filled with examples of bifurcations, the concept of bifurcation
as a change in the topological equivalence class is never formulated.
Similarly, Peixoto's theorem on two-dimensional flows is not stated.
The discussion of elementary bifurcations for maps and vector fields
omits center manifolds and normal forms entirely, so the reader cannot
appreciate the full significance of the simple one- and
two-dimensional examples. Hopf bifurcation in maps is not presented.
The importance of homoclinic points is stressed, but Melnikov methods
for detecting them are not mentioned. In a discussion of a damped
driven pendulum, the main point of which is to introduce a return map
with fractal basin boundaries, the authors miss an opportunity to
explain how dimensional analysis allows a model to be simplified.
Instead we are told that to obtain the dimensionless model, it is
necessary to "use a pendulum of length l = g." How to achieve this
feat is presumably an exercise for the reader!

I think the text
should be quite useful for physics graduate courses and honors-level
undergraduate courses, although it contains far more material than
could be covered in a single term. By the same token, it is a serious
introduction, with wider coverage of the subject than is readily
available from any other single source.